Average Error: 59.5 → 59.5
Time: 14.1s
Precision: binary64
\[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
↓
\[\begin{array}{l} t_0 := \sqrt{\cos x}\\ t_1 := \sqrt[3]{\log \left(\left(e^{x}\right) \bmod t_0\right) - x}\\ {\left({e}^{\left(t_1 \cdot t_1\right)}\right)}^{\left(\sqrt[3]{\log \left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{t_0}^{3}}\right)\right)}{e^{x}}\right)}\right)} \end{array} \]
\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}
↓
\begin{array}{l}
t_0 := \sqrt{\cos x}\\
t_1 := \sqrt[3]{\log \left(\left(e^{x}\right) \bmod t_0\right) - x}\\
{\left({e}^{\left(t_1 \cdot t_1\right)}\right)}^{\left(\sqrt[3]{\log \left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{t_0}^{3}}\right)\right)}{e^{x}}\right)}\right)}
\end{array}

(FPCore (x) :precision binary64 (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (cos x))) (t_1 (cbrt (- (log (fmod (exp x) t_0)) x))))
   (pow
    (pow E (* t_1 t_1))
    (cbrt (log (/ (fmod (exp x) (cbrt (pow t_0 3.0))) (exp x)))))))
double code(double x) {
	return fmod(exp(x), sqrt(cos(x))) * exp(-x);
}
↓
double code(double x) {
	double t_0 = sqrt(cos(x));
	double t_1 = cbrt((log(fmod(exp(x), t_0)) - x));
	return pow(pow(((double) M_E), (t_1 * t_1)), cbrt(log((fmod(exp(x), cbrt(pow(t_0, 3.0))) / exp(x)))));
}

Error

Bits error versus x

Derivation

  1. Initial program 59.5

    \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

  2. Simplified59.5

    \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}} \]

  3. Applied add-exp-log_binary6459.5

    \[\leadsto \color{blue}{e^{\log \left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)}} \]

  4. Applied pow1_binary6459.5

    \[\leadsto e^{\log \color{blue}{\left({\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)}^{1}\right)}} \]

  5. Applied log-pow_binary6459.5

    \[\leadsto e^{\color{blue}{1 \cdot \log \left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)}} \]

  6. Applied exp-prod_binary6459.5

    \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\log \left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)}} \]

  7. Simplified59.5

    \[\leadsto {\color{blue}{e}}^{\log \left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)} \]

  8. Applied add-cbrt-cube_binary6459.5

    \[\leadsto {e}^{\log \left(\frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\sqrt[3]{\left(\sqrt{\cos x} \cdot \sqrt{\cos x}\right) \cdot \sqrt{\cos x}}\right)}\right)}{e^{x}}\right)} \]

  9. Simplified59.5

    \[\leadsto {e}^{\log \left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{\color{blue}{{\left(\sqrt{\cos x}\right)}^{3}}}\right)\right)}{e^{x}}\right)} \]

  10. Applied add-cube-cbrt_binary6459.5

    \[\leadsto {e}^{\color{blue}{\left(\left(\sqrt[3]{\log \left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\left(\sqrt{\cos x}\right)}^{3}}\right)\right)}{e^{x}}\right)} \cdot \sqrt[3]{\log \left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\left(\sqrt{\cos x}\right)}^{3}}\right)\right)}{e^{x}}\right)}\right) \cdot \sqrt[3]{\log \left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\left(\sqrt{\cos x}\right)}^{3}}\right)\right)}{e^{x}}\right)}\right)}} \]

  11. Applied pow-unpow_binary6459.5

    \[\leadsto \color{blue}{{\left({e}^{\left(\sqrt[3]{\log \left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\left(\sqrt{\cos x}\right)}^{3}}\right)\right)}{e^{x}}\right)} \cdot \sqrt[3]{\log \left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\left(\sqrt{\cos x}\right)}^{3}}\right)\right)}{e^{x}}\right)}\right)}\right)}^{\left(\sqrt[3]{\log \left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\left(\sqrt{\cos x}\right)}^{3}}\right)\right)}{e^{x}}\right)}\right)}} \]

  12. Simplified59.5

    \[\leadsto {\color{blue}{\left({e}^{\left(\sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x} \cdot \sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x}\right)}\right)}}^{\left(\sqrt[3]{\log \left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\left(\sqrt{\cos x}\right)}^{3}}\right)\right)}{e^{x}}\right)}\right)} \]

  13. Final simplification59.5

    \[\leadsto {\left({e}^{\left(\sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x} \cdot \sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x}\right)}\right)}^{\left(\sqrt[3]{\log \left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\left(\sqrt{\cos x}\right)}^{3}}\right)\right)}{e^{x}}\right)}\right)} \]

Reproduce

herbie shell --seed 2022125 
(FPCore (x)
  :name "expfmod"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))